Directional Tunneling Escape from Nearly Spherical Optical Resonators

We report the surprising observation of directional tunneling escape from nearly spherical fused-silica optical resonators, in which most of the phase space is filled with nonchaotic regular trajectories. Experimental and theoretical studies of the dependence of the far-field emission pattern on both the degree of deformation and the excitation condition show that nonperturbative phase-space structures in the internal ray dynamics profoundly affect tunneling leakage of the whispering-gallery modes.

Figure 1

(a)–(d): Far-field emission patterns of WG modes. Insets: bottom view of the resonators showing the progression of shapes in the $x$-$y$ cross section, from which we determine $ϵ=6.7\%$ (a), $ϵ=2.2\%$ (b), and $ϵ=1.2\%$ (c),(d). WG modes were launched at $\sin {\chi }_0\approx 1$ in (a)–(c), and at $\sin {\chi }_0\approx 0.8$ in (d); $\chi$ is the internal angle of incidence with respect to the surface normal. (e)–(h): the spectra corresponding to the modes in (a)–(d), from which we deduce the $Q$ factors. Inset to (d): stable (solid) and unstable (dashed) 4-bounce orbits in a quadrupole at $ϵ=1.2\%$.

Figure 2

Top: Intensity pattern of WG modes with parameters $kR=112.063$, $ϵ=6.5\%$ (a) and $kR=112.452$, $ϵ=3.4\%$ (b). The $Q$ factors are $4\times {10}^4$ (a) and $2\times {10}^5$ (b), and the overall directionality agrees with that in Fig. 1b, 1d, respectively. The values of $ϵ$ differ from Fig. 1 because the simulation approximates the unknown details of the experimental shape by a pure quadrupole [12]. Bottom: solid histogram in (c) shows the angular-momentum distribution of the mode in (b). A very high-$Q$ WG mode at the same deformation [(c), dashed], with $kR=113.229$ and $Q=3\times {10}^{14}$ radiates predominantly into $\phi =90°$, as shown in the far-field pattern (d).